No. Google as a verb should not be capitalized. Because if you put 'G' capital, you mean the word 'Google' as a company (proper noun). You cannot company something.
I found this on Wikipedia. It's useful.
The first recorded usage of google used as a participle, thus supposing an intransitive verb, was on July 8, 1998, by Google co-founder Larry Page himself, who wrote on a mailing list: "Have fun and keep googling!"
Its earliest known use (as a transitive verb) on American television was in the "Help" episode of Buffy the Vampire Slayer (October 15, 2002), when Willow asked Buffy, "Have you googled her yet?"
There's no first letter capital.
OALD has an entry of the verb google.
It's observed that when the tool of doing something becomes too popular (and omnipotent in its field!), its proper noun, over the period of time, becomes a verb and then the first capital letter is lowercased.
Another such proper nouns are --Photoshop, when used as a verb becomes photoshop (no capital) and Xerox, when used as a verb becomes xerox (no capital again).
On one blog (as fluffy says) it's written:
The verb "google" (no capital) has been used in the recent years with the meaning "search (something) on the web", even though you use other search engine than Google (capital).
[Generally, a verb is not capitalized. However, the only verb with first letter capital I have come across is Christianize. I had asked that question here on this board some time ago.]
Noun and verb usage:
Did you google this term? Vs. Did you search this term on Google?
You can photoshop this image to make it better Vs. You can make this image better using Photoshop.